Is it possible to find the nodes connected to a node in python?
The python path to the links between each node is

bpy.data.materials[].node_tree.nodes[].inputs[].links[].from_node


So for each material we want to start at the Material Output node then loop through each array of inputs[] and links[] to move through the connections. Going this way (using from_node) there should only be one link for each input, but if you want to reverse the order (by using to_node) you can have multiple links for each output.

import bpy

def followLinks(node_in):
    for n_inputs in node_in.inputs:
        for node_links in n_inputs.links:
            print("going to " + node_links.from_node.name)
            followLinks(node_links.from_node)

for mat in bpy.data.materials:
    print("Traversing " + mat.name)
    for mat_node in mat.node_tree.nodes:
        if mat_node.type == 'OUTPUT_MATERIAL':
            # we start at the material output node
            print("Starting at " + mat_node.name)
            followLinks(mat_node)